$ A = \left[\begin{array}{rr}1 & -2 \\ 2 & 3 \\ 1 & -1\end{array}\right]$ $ w = \left[\begin{array}{r}2 \\ 1\end{array}\right]$ What is $ A w$ ?
Solution: Because $ A$ has dimensions $(3\times2)$ and $ w$ has dimensions $(2\times1)$ , the answer matrix will have dimensions $(3\times1)$ $ A w = \left[\begin{array}{rr}{1} & {-2} \\ {2} & {3} \\ \color{gray}{1} & \color{gray}{-1}\end{array}\right] \left[\begin{array}{r}{2} \\ {1}\end{array}\right] = \left[\begin{array}{r}? \\ ? \\ ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ w$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ w$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ w$ , and so on. Add the products together. $ \left[\begin{array}{r}{1}\cdot{2}+{-2}\cdot{1} \\ ? \\ ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ w$ and add the products together. $ \left[\begin{array}{r}{1}\cdot{2}+{-2}\cdot{1} \\ {2}\cdot{2}+{3}\cdot{1} \\ ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{r}{1}\cdot{2}+{-2}\cdot{1} \\ {2}\cdot{2}+{3}\cdot{1} \\ \color{gray}{1}\cdot{2}+\color{gray}{-1}\cdot{1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{r}0 \\ 7 \\ 1\end{array}\right] $